Can you elaborate more on the derivation of $d^{\text{VCM}}_1$ for $\mathbf{z}_1$ in equation (31)? And I found the intuitive definition of "vertex merging" in section 2.1 to be a bit confused. The original paper says that the PDFs of the same-length paths sampled by BDPT and PM should be expressed w.r.t. the same measure thus MIS can be applied. But now this intuition use paths of different length which seems to render the MIS invalid?